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Shaping the PDF of the state variable based on piecewise linear control for non-linear stochastic systems

非线性随机系统中基于分段线性控制的PDF形状控制研究

Abstract

In this paper, we propose a shape-control scheme for non-linear stochastic dynamical systems to enable us to shape the probability density function (PDF) of the state variable. First, we derive the PDF analytical expression using the Fokker-Planck-Kolmogorov (FPK) equation obtained from stochastic systems. Then, we control the PDF shape by devising a piecewise linear control law whose parameters are calculated using the conjugated gradient method. Finally, we perform contrast simulation experiments to validate the effectiveness and superiority of the proposed algorithm.

概要

创新点

与传统的均值和方差控制方法相比, 本文提出了一种状态变量的概率密度函数(Probability Density Function, 简称PDF)形状控制方法。我们的研究对象是非线性随机系统, 并且用PDF形状逼近作为性能指标。在过去的PDF形状控制研究中, 由于多项式的灵活性和连续性, 我们一直采用多项式形式作为控制规律, 但这明显增加了计算量和算法的复杂度. 本文设计了一个分段线性控制规律实现PDF形状的控制, 线性控制规律只包含两个系数和一个分段点, 在一定范围内呈现线性特性, 这使得整个算法变得更加简单, 在程序实现时节省运行时间. 线性分段控制尽管已经在文献[22]中研究过, 但本文有三点不同之处: 1、文献中的控制目标是没有考虑测量噪声的的输出PDF, 而本文的控制目标是状态变量的PDF; 2、文献中地稳态PDF是近似的, 而我们解出了一个精确的稳态PDF; 3、尽管我们采用相同的控制结构, 但获取控制器增益的方法是不同的, 我们采用的是共轭梯度法得到了控制器的增益. 对比实验结果表明, 所提出的分段线性控制方法可以有效地控制状态变量的PDF形状, 并且当期望的PDF形状是不对称双峰时, 其控制效果要优于非线性多项式控制. 同时, 不管期望的PDF形状如何, 线性控制方法的运行时间都要比非线性控制方法的短. 仿真实例中的非线性系统是一个强非线性系统, 这说明所提出的控制算法对强非线性系统是有效的.

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Correspondence to Fucai Qian.

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Wang, L., Qian, F. Shaping the PDF of the state variable based on piecewise linear control for non-linear stochastic systems. Sci. China Inf. Sci. 59, 112207 (2016). https://doi.org/10.1007/s11432-015-0401-9

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Keywords

  • non-linear stochastic systems
  • probability density function
  • shape control
  • piecewise linear control
  • Fokker-Planck-Kolmogorov (FPK) equation

关键词

  • 非线性随机系统
  • 概率密度函数
  • 形状控制
  • 分段线性控制
  • FPK(Fokker-Planck-Kolmogorov)方程